##
**On Poonen’s conjecture concerning rational preperiodic points of quadratic maps.**
*(English)*
Zbl 1316.37042

This note presents some evidence for B. Poonen’s conjecture [Math. Z. 228, No. 1, 11–29 (1998; Zbl 0902.11025)]: There is no \(c\in \mathbb Q\) such that \(\phi_c(z) = z^2 + c\) has a \(\mathbb Q\)-rational periodic point with exact period greater than \(3\). The authors check this for \(h(c) \leq 8\log 10\) in Proposition 1, where \(h\) is the standard logarithmic height function on \(\mathbb Q\). In Proposition 2, they give an analogous result for a quadratic field \(K\) with discriminant in \([-4000,4000]\): no \(K\)-rational periodic point of exact period greater than \(6\) if \(h(c) \leq 3\log 10\). The method is standard and the same as the computation of torsions on elliptic curves, namely modding out by good-reduction primes and relating the period over finite fields to the period over number fields. The authors reduce the amount of computations by Lemmas 2 and 3, in which they show that a certain ideal has to be a square if there is even a single periodic point in \(K\) for \(\phi_c\). Many reduction-types modulo small primes can also be ruled out, for example if they just have superattracting cycles of small periods. The authors comment that while considering more primes reduces the proportion of reduction-types to check, this increases the amount of computations for each \(c\). They also show in Proposition 4 that the largest exact period of a \(K\)-rational periodic point grows at least linearly in \([K:\mathbb Q]\).

Reviewer: Yu Yasufuku (Tokyo)

### MSC:

37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |

37P35 | Arithmetic properties of periodic points |

### Keywords:

uniform boundedness; periodic points; quadratic polynomials; Poonen’s conjecture; Morton-Silverman conjecture### Citations:

Zbl 0902.11025
PDF
BibTeX
XML
Cite

\textit{B. Hutz} and \textit{P. Ingram}, Rocky Mt. J. Math. 43, No. 1, 193--204 (2013; Zbl 1316.37042)

### References:

[1] | E.V. Flynn, B. Poonen and E. Schaefer, Cycles of quadratic polynomials and rational points on a genus \(2\) curve , Duke Math. J. 90 (1997), 435-463. · Zbl 0958.11024 |

[2] | S. Kamienny, Torsion points on elliptic curves , Bull. Amer. Math. Soc. 23 (1990), 371-373. · Zbl 0714.11033 |

[3] | M. Manes, \({\mathbf Q}\)-rational cycles for degree-\(2\) rational maps having an automorphism , Proc. Lond. Math. Soc. 96 (2008), 669-696. · Zbl 1213.14048 |

[4] | B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33-186. · Zbl 0394.14008 |

[5] | L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), 437-449. · Zbl 0936.11037 |

[6] | P. Morton, Arithmetic properties of periodic points of quadratic maps , II, Acta Arith. 87 (1998), 89-102. · Zbl 1029.12002 |

[7] | P. Morton and J. Silverman, Periodic points, multiplicities, and dynamical units , J. reine angew. Math. 461 (1995), 81-122. · Zbl 0813.11059 |

[8] | B. Poonen, The classification of rational preperiodic points of quadratic polynomials over \({\mathbf Q}\) : A refined conjecture , Math. Z. 228 (1998), 11-29. · Zbl 0902.11025 |

[9] | J.H. Silverman, The arithmetic of dynamical systems , Grad. Texts Math. 241 , Springer-Verlag, New York, 2007. · Zbl 1130.37001 |

[10] | M. Stoll, Rational \(6\)-cycles under iteration of quadratic polynomials , Lond. Math. Soc. J. Comput. Math. 11 (2008), 367-380. · Zbl 1222.11083 |

[11] | R. Walde and P. Russo, Rational periodic points of the quadratic function \(Q_c=x^2+c\) , The Amer. Math. Month. 101 (1994), 318-331. \noindentstyle · Zbl 0804.58036 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.